Hyperbolic Distributions in Social Science Zipf’s law and invariants of employment dynamics

1. Hyperbolic Distributions in Social ScienceZipf’s law and invariants of employment dynamics Bernd Schmeikal Wiener Institute for Social Science Documentation and Methodology (WISDOM) 10th International Conference of Numerical Analysis and Applied Mathematics, 19-25 September, 2012, Kypriotis Hotels and Conference Center,Kos, Greece

2. social space • refering to Bourdieu‘s concept we regard social space as not metric • categorial multidimensional • pointfree • in a non metric domain, eventually partially ordered. • is there a mathematics for such undertaking ?

3. Whiteheads Extensionskalkül • »inclusion-based pointfree geometry« by posets based on a complete partial order • reflexive • transitive • antisymmetrisch • no extremes • directed • proper parts rule

4. completeness • dense • Robinson und Zakon (1960, theorem 2.3) • Only in a dense strict partial order without extreme locations, is the system complete • In our case it is not known what that means for the arithmetic procedures.

5. Austrian labor market • an old time-series decomposed • approximating equation of motion

6. timeseries up to date

7. this raises questions • Can we describe mechanisms that bring about the time series? • Are there specific locations in social space which give rise to peculiar contributions to the number of workless?

8. gender

9. Heinz and two kinds of statistics Empirical quantities such as body height, weight of people, length of nails a. s. o. cluster around a mean value. Considering 4 inch nails, the data set is indeed observed to fit a normal distribution. But this holds only for a certain type of nail. If we consider all nails from length, say, 10 mm to 1000 mm, we do not get a normal distribution, but a power law. Heinz von Foerster had pointed out in a discussion of Zipf’s law at the Macy meeting 1952: “ … I think the difference in the two kinds of statistics is that here we are dealing with a number of different kinds, whereas in the other case in the Gaussian situation, for instance, - we are dealing with one kind only.”

10. coastline paradox To comprehend this, let us consider the coastline paradox found by Lewis Fry Richardson. He observed that the coastline of a landmass does not have a well-defined length. The length of the coastline depends on the rulers used to measure it. Since a landmass can be measured at all scales, from hundreds of kilometers in size to fractions of a millimeter, there is in principle no limit to the size of the smallest feature that should not be measured around. Hence no definite perimeter to the landmass can be achieved.

11. niche-size & labor market share When we measure size distributions of labor market shares, we need a measure analogous to the length scale used in the coastline paradox. We are used that frequency-distributions are given by absolute or relative frequencies or briefly ‘number of people’. For example we count the ‘number of scientists’ in a special field of science. These numbers characterize the distribution. But in the RISC-approach the situation is precisely reverse: These numbers now provide us the ‘measure’ for the phenomena we want to measure. We call it the ‘niche-size’. Niche-size is the measure of a labor market share, and, in a way, it parallels the ruler with which we measure the coastline.

12. count the niches of given size The counts that give us the distribution, however, are given by the number of such niches existing in the market. The measure in our RISC-approach is given by the occupation number of a multivariate categorical profile. This is a natural number. So the surveyed metric magnitudes are equal to the sizes of niches in the social space. The social space is a labor market.

13. data We use the labor-service database ‘AMS-BMASK’ and two very large empirical datasets involving fourteen social categorial variables measured in the whole population in order to demonstrate the relevance of the RISC approach and the validity of the statistical model developed. Selecting gender, age-group, federal state, nation, occupational group, we allow for 8712 combinations which constitute the locations in social space. Each location has a measure given by the size of the niche.

14. multivariate modal profile • Definition 1: the ‘multivariate modal profile’ denotes that categorical combination where there is a maximum number of counts. The multivariate modal profile in social space signifies its largest niche. • Definition 2: the ‘measure’ in a RISC-frame of labor force sociology is given by the occupation number, that is, the absolute frequency of persons counted at some multivariate attribute profile. It is a natural number. The metric magnitudes are now sizes of social niches.

15. niche distribution of dependent employees

16. Zipf distribution • we have 8712 multivariate profiles • omitting the empty locations, we count 105 niches with occupation number = magnitude • multivariate modal profile has magnitude 19294 • stochstic varibale has a density function with , and the constant c essentially given by the Riemann zeta function, that is, . In the year 2001 we obtain and . The following figure shows the approximation.

17. Zipf distribution of social locations

18. the largest DW locations in 2001 • 01/01; Male, 40-44, Vienna, Aut, Publ Admin, dependent worker [19395] • 01/02; Male, 40-44, Vienna, Aut, Publ Admin, dependent worker [19417] • 01/03; Male, 40-44, Vienna, Aut, Publ Admin, dependent worker [19535] • 01/04; Male, 40-44, Vienna, Aut, Publ Admin, dependent worker [19606] • 01/05; Male, 40-44, Vienna, Aut, Publ Admin, dependent worker [19395] • 01/06; Male, 40-44, Vienna, Aut, Publ Admin, dependent worker [19700] • 01/07; Male, 40-44, Vienna, Aut, Publ Admin, dependent worker [18955]

19. the largest WL locations in 2001 until July 01/01; Male, 40-44, Stmk, Aut, Building trade, workless [1983] 01/02; Male, 40-44, Stmk, Aut, Building trade, workless, [1739] 01/03; Male, 40-44, Stmk, Aut, Building trade, workless, [1010] 01/04; Male, 60-64, Vienna, Aut, Trade+mot vehicle ind+Rep, workless, [889] 01/05; Male, 60-64, Vienna, Aut, Trade+mot vehicle ind+Rep, workless, [878] 01/06; Male, 60-64, Vienna, Aut, Trade+mot vehicle ind+Rep, workless, [862]

20. the largest WL locations in 2001 July-December 01/07; female, 40-44, Vienna, Aut, Trade+mot vehicle ind+Rep, workless [905] 01/08; female, 40-44, Vienna, Aut, Trade+mot vehicle ind+Rep, workless [921] 01/09; female, 35-39, Vienna, Aut, Trade+mot vehicle ind+Rep, workless [895] 01/10; female, 35-39, Vienna, Aut, Trade+mot vehicle ind+Rep, workless [909] 01/11; female, 40-44, Vienna, Aut, Trade+mot vehicle ind+Rep, workless [893] 01/12; Male, 40-44, Stmk, Aut, Building trade, workless [1552]

21. in the critical year 2008

22. accumulated frequency • Integrating the density function we get a first rough approximation of the cumulative distribution • In practise we estimate and in large amounts of data such as AMS, using SPSS as follows. Letting unknown, we take the occupation number as argument for the cumulative distribution function and assume a power law with the familiar SPSS-constants and . Estimating these parameters we get the equations • and ‘fractal dimension’ .

23. SPSS estimate of significance

24. facts transcending the pure power law

25. new niches • There is an increasing number of niches for marginally employed workers. Most of those concern the female working population. • There is a decreasing industry and an increasing, highly unstable knowledge sector and so on. • The phenomenology of superimposing processes is not easily understood.

26. Zipf‘s law at extreme ends

27. Literatur • WISDOM-FORSCHUNG, Zwischenbericht/Interim Report Nr. 47 Regionalentwicklung in Mittel- und Osteuropa: Szenarien für Beschäftigung, Qualifikationen und Migrationsbewegungen Teil I: Übersichten und Zusammenfassungen Peter Fleissner, Karl H. Müller (Hrsg.), Bernd Schmeikal, Michael Schreiber et al., Mai 2011 FÖRDERGEBER: Mag. Richard Fuchsbichler, MBA Gruppenleiter Gruppe S Bundesministerium für Arbeit, Soziales und Konsumentenschutz (BMASK) Sektion VI / Abteilung VI/6

28. Beck, U.; Risikogesellschaft. Auf dem Weg in eine andere Moderne. Suhrkamp, Frankfurt a.M. 1986. • Box, G. E. P., Jenkins, G.M.; (1976). Time Series Analysis. (Rev.Ed). San Francisco. (First edn., 1970). • Yanguang Chen, Zipf‘s law, noise, and fractal hierarchy Department of Geography, College of Urban and Environmental Sciences, Peking University, 100871, Beijing, China. Email: chenyg@pku.edu.cn • Yanguang Chen, Scaling Separation and Reconstruction of Zipf’s law, http://arxiv.org/ftp/arxiv/papers/1104/1104.3199.pdf • Kajfez-Bogataj, L, Müller, K., Svetlik, I., Tos, N.; Modern RISC Societies. Edition echoraum, Vienna 2010, p. 121ff.

29. Lotka, A. J.; (1926). The frequency distribution of scientific productivity. Journal of the Washington Academy of Sciences, 16: 317-323. • Mandelbrot, B.; Fractals: Form, Chance and Dimension. W H Freeman and Co, New York 1977. • Richardson L. F.; Weather Prediction by Numerical Process. Cambridge University Press, London 1922. • F. Richardson, L. F.; (1961). The problem of contiguity: An appendix to Statistic of Deadly Quarrels.General systems: Yearbook of the Society for the Advancement of General Systems Theory. (Ann Arbor, Mich.: The Society, [1956-: Society for General Systems Research) 6 (139): 139–187.